1
JEE Main 2023 (Online) 12th April Morning Shift
Numerical
+4
-1

Let $$[x]$$ be the greatest integer $$\leq x$$. Then the number of points in the interval $$(-2,1)$$, where the function $$f(x)=|[x]|+\sqrt{x-[x]}$$ is discontinuous, is ___________.

2
JEE Main 2023 (Online) 10th April Morning Shift
Numerical
+4
-1

Let $$f:( - 2,2) \to R$$ be defined by $$f(x) = \left\{ {\matrix{ {x[x],} & { - 2 < x < 0} \cr {(x - 1)[x],} & {0 \le x \le 2} \cr } } \right.$$ where $$[x]$$ denotes the greatest integer function. If m and n respectively are the number of points in $$( - 2,2)$$ at which $$y = |f(x)|$$ is not continuous and not differentiable, then $$m + n$$ is equal to ____________.

3
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1

Let $$\mathrm{k}$$ and $$\mathrm{m}$$ be positive real numbers such that the function $$f(x)=\left\{\begin{array}{cc}3 x^{2}+k \sqrt{x+1}, & 0 < x < 1 \\ m x^{2}+k^{2}, & x \geq 1\end{array}\right.$$ is differentiable for all $$x > 0$$. Then $$\frac{8 f^{\prime}(8)}{f^{\prime}\left(\frac{1}{8}\right)}$$ is equal to ____________.

4
JEE Main 2023 (Online) 6th April Morning Shift
Numerical
+4
-1

Let $$a \in \mathbb{Z}$$ and $$[\mathrm{t}]$$ be the greatest integer $$\leq \mathrm{t}$$. Then the number of points, where the function $$f(x)=[a+13 \sin x], x \in(0, \pi)$$ is not differentiable, is __________.